Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Average Error: 11.2 → 0.8

Time: 7.7s

Precision: binary64

Cost: 6984

Math

\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 250000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -3.25e+74)
   (/ (exp (- y)) x)
   (if (<= x 250000000000.0) (/ 1.0 x) (/ (/ 1.0 x) (exp y)))))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

↓

double code(double x, double y) {
	double tmp;
	if (x <= -3.25e+74) {
		tmp = exp(-y) / x;
	} else if (x <= 250000000000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / x) / exp(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.25d+74)) then
        tmp = exp(-y) / x
    else if (x <= 250000000000.0d0) then
        tmp = 1.0d0 / x
    else
        tmp = (1.0d0 / x) / exp(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

↓

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.25e+74) {
		tmp = Math.exp(-y) / x;
	} else if (x <= 250000000000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = (1.0 / x) / Math.exp(y);
	}
	return tmp;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

↓

def code(x, y):
	tmp = 0
	if x <= -3.25e+74:
		tmp = math.exp(-y) / x
	elif x <= 250000000000.0:
		tmp = 1.0 / x
	else:
		tmp = (1.0 / x) / math.exp(y)
	return tmp

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

↓

function code(x, y)
	tmp = 0.0
	if (x <= -3.25e+74)
		tmp = Float64(exp(Float64(-y)) / x);
	elseif (x <= 250000000000.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(Float64(1.0 / x) / exp(y));
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.25e+74)
		tmp = exp(-y) / x;
	elseif (x <= 250000000000.0)
		tmp = 1.0 / x;
	else
		tmp = (1.0 / x) / exp(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[x, -3.25e+74], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 250000000000.0], N[(1.0 / x), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[Exp[y], $MachinePrecision]), $MachinePrecision]]]

Target

Original	11.2
Target	8.0
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < -3.24999999999999981e74

   1. Initial program 14.5

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]

   2. Simplified 14.5

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]

   3. Taylor expanded in x around inf 0.1

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{e^{-y}}{x}} \]

   if -3.24999999999999981e74 < x < 2.5e11

   1. Initial program 10.4

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]

   2. Simplified 10.4

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]

   3. Taylor expanded in x around 0 1.5

      \[\leadsto \color{blue}{\frac{1}{x}} \]

   if 2.5e11 < x

   1. Initial program 10.6

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]

   2. Simplified 10.6

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]

   3. Taylor expanded in x around inf 0.1

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}} \]

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{e^{-y}}{x}} \]

   5. Applied egg-rr 0.1

      \[\leadsto \color{blue}{{\left(x \cdot e^{y}\right)}^{-1}} \]

   6. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{e^{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 250000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{e^{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	6920

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -3.25 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 250000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq 1.8:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+208}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
Error	52.8
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1.0 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))